Finite Group Databasebeta

← #6514010

Subgroups — #6514010

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34979C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#27681C212yesmaximal, complemented
4#27647C212yesmaximal, complemented
5#27631C212yesmaximal, complemented
6#27631C212yesmaximal, complemented
7#26322C212yesmaximal, complemented
8#26296C212yesmaximal, complemented
9#26278C212yesmaximal, complemented
10#26234C212yesmaximal, complemented
11#26216C214yesmaximal, complemented
12#26144C214yesmaximal, complemented
13#21894C211yesmaximal, characteristic, complemented
14#21860C211yesmaximal, characteristic, complemented
15#21849C211yesmaximal, characteristic
16#21362C211yesmaximal, characteristic, complemented
17#21171C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#483C2214yes—
22C2×#431C2211yescharacteristic
23C2×#428C2212yes—
24C2×#423C2211yescharacteristic
25#2611—22no—
26#2568C2211yescharacteristic
27#2547C2211yescharacteristic
28#2546C2211yescharacteristic
29#2542C2211yescharacteristic
30#2499—21no—
31#2432—21no—
32#2415C2212yes—
33#2415C2212yes—
34#2395—22no—
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4C2212yes—
37D4×C4⋊C4—22no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2214yes—
40D4×C4⋊C4C2212yes—
41#2317—22no—
42#2317—21no—
43C4×C4⋊Q8—21no—
44C4×#124—21no—
45C4×C22⋊Q8C2212yes—
46C4×C22⋊Q8C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2212yes—
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2095C2214yes—
54#2084C2214yes—
55#2075C2214yes—
56#2075C2214yes—
57#2072C2214yes—
58#2035C2214yes—
59#2035C2214yes—
60#1990C2212yes—
61#1988C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1956C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1948C2211yescharacteristic
68#1948C2212yes—
69#1948C2212yes—
70#1947C2211yescharacteristic
71#1947C2212yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1928C2214yes—
75#1928C2214yes—
76#1919C2214yes—
77#1919C2214yes—
78#1894C2214yes—
79#1872C2214yes—
80#1872C2214yes—
81#1870C2214yes—
82#1854C2212yes—
83#1854C2212yes—
84#1852C2214yes—
85#1823C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×#106C2212yes—
94#1736C2212yes—
95C22×#41D411yescharacteristic
96C22×M16C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—21no—
103C2×#124—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108#520—48no—
109C4×#41—24no—
110C4×#41—24no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—24no—
117C2×C4×D4—22no—
118C2×C4×D4D412yes—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×#117—22no—
123C2×#117—21no—
124C2×#117—21no—
125C2×#117D412yes—
126C2×#117D411yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#483—48no—
138#431—24no—
139#428—44no—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#106C2312yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146#423—24no—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C4×C8C2312yes—
159#400—22no—
160#399—22no—
161#398—22no—
162#397—22no—
163#397—22no—
164#397—22no—
165#394—22no—
166#394—22no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391C2312yes—
171#391C2312yes—
172#390—21no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—24no—
177#389—22no—
178#389—24no—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—24no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—22no—
192#384—21no—
193#384—22no—
194#384—21no—
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2314yes—
199#384C2312yes—
200#382—21no—
201#382—22no—
202#382—21no—
203#380—22no—
204#380—21no—
205#380C2312yes—
206#380C2314yes—
207#380C2312yes—
208#380C2312yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223#376—21no—
224C43—21no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—24no—
240C22×Q8—21no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×D4C2×D412yes—
244C22×D4C2×D411yescharacteristic
245C23×C4—22no—
246C23×C4—24no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—42no—
255C2×M16—42no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—44no—
264#125—44no—
265#124—44no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C22⋊Q8—44no—
273C4×Q8—24no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—48no—
279C4×Q8—44no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—416no—
287C4×D4—48no—
288#117—24no—
289#117—28no—
290#117—22no—
291#117—22no—
292#117—41no—
293#117—41no—
294#117—48no—
295#117—44no—
296#117—44no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4—84no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42#4114yes—
366C2×C42#4114yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—42no—
371#107—42no—
372#107—44no—
373#106—28no—
374#106—24no—
375#106—24no—
376C4⋊C8—24no—
377C4⋊C8—24no—
378C4⋊C8—22no—
379C4⋊C8—22no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384C22⋊C8—24no—
385C22⋊C8—44no—
386C22⋊C8—44no—
387C4×C8—24no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—22no—
404C24C22×D411yescharacteristic
405C24C22×D412yes—
406#41—232no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8#13714yes—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4AGL(1,ℤ/8)14yes—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—84no—
457C22×C4—82no—
458C22×C4—84no—
459C22×C4—82no—
460C22×C4#12314yes—
461C22×C4#12314yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473M16—48no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—82no—
519C4⋊C4—84no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4H(5,𝔽2)12yes—
523C4⋊C42−1+412yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—42no—
541C42—42no—
542C42—48no—
543C42—44no—
544C42—44no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42#12314yes—
550C42#12314yes—
551C42C2×#4112yes—
552C42C2×#4112yes—
553C42C2×#4112yes—
554C42C2×#4112yes—
555C23—22no—
556C23—22no—
557C23—21no—
558C23—22no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23C2×#12312yes—
568C23C2×AGL(1,ℤ/8)14yes—
569C23C23×D411yescharacteristic
570Q8—28no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—84no—
576D4—216no—
577D4—28no—
578C2×C4—216no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—168no—
632C2×C4—164no—
633C2×C4C2×#12312yes—
634C2×C4C2×#12312yes—
635C2×C4C2×#12312yes—
636C2×C4C2×#12312yes—
637C2×C4C2×#12312yes—
638C2×C4C2×AGL(1,ℤ/8)12yes—
639C2×C4C2×#13714yes—
640C2×C4C2×#13712yes—
641C2×C4#57712yes—
642C2×C4#57812yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×2−1+411yescharacteristic
647C2×C4#58511yescharacteristic
648C2×C4#58511yescharacteristic
649C8—44no—
650C8—42no—
651C8—42no—
652C22—24no—
653C22—22no—
654C22—24no—
655C22—22no—
656C22—24no—
657C22—22no—
658C22—41no—
659C22—42no—
660C22—41no—
661C22—42no—
662C22—41no—
663C22—42no—
664C22—42no—
665C22—41no—
666C22C22×#12311yescharacteristic
667C22#348811yescharacteristic
668C22#348911yescharacteristic
669C22C22×AGL(1,ℤ/8)11yescharacteristic
670C22C22×#13711yescharacteristic
671C22C2×#57711yescharacteristic
672C22C2×#57811yescharacteristic
673C4—22no—
674C4—24no—
675C4—22no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—81no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4—81no—
686C4—82no—
687C4—82no—
688C4#309214yes—
689C4#309714yes—
690C4#350611yescharacteristic
691C4#350711yescharacteristic
692C4#351111yescharacteristic
693C4#351211yescharacteristic
694C2—44no—
695C2—42no—
696C2#3551711yescharacteristic
697C2C2×#309211yescharacteristic
698C2C2×#309711yescharacteristic
699C2#6198311yescharacteristic
700C2#6200011yescharacteristic
701C2#6213211yescharacteristic
702C2#6214711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #27681C2semidirect
4. #27647C2semidirect
5. #27631C2semidirect
6. #27631C2semidirect
7. #26322C2semidirect
8. #26296C2semidirect
9. #26278C2semidirect
10. #26234C2semidirect
11. #26216C2semidirect
12. #26144C2semidirect
13. #21894C2semidirect
14. #21860C2semidirect
16. #21362C2semidirect
17. #21171C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#349792
#341832
#276812
#276472
#276312
#263222
#262962
#262782
#262342
#262162
#261442
#218942
#218602
#218492
#213622
#211712